Deriving the Boltzmann formula, defining temperature, and simulating liquid/vapor.
@SpectralCollective has the second part: https://youtu.be/yEcysu5xZH0
You can play with a simulation of this model here: https://vilas.us/simulations/liquidvapor/
These lessons are funded directly by viewers: https://3b1b.co/support
Home page: https://www.3blue1brown.com

Notes from Vilas:
1) This open problem is to prove the ergodicity of the deterministic dynamical systems that are used to model the molecule-level physics. A good example of such a dynamical system is the box with particles evolving according to Newton's laws with elastic collisions, like in the video.

2) This video assumes that all probability distributions are discrete, which is the case in the simulations. But one can also set up this formalism for systems with continuous state spaces. Again, a good example is particles in a box, which are not restricted to a lattice, which is only used for visualization purposes in this video.

3) Strictly speaking, these "derivatives" don't make sense since in our simulations the energy can only take on a discrete set of values. But a derivative is the right way to think about this, and is the correct notion in a limiting sense.

4) The factor of -T in the definition of the chemical potential is sort of a historical leftover, but including it has the convenient side effect of allowing the same Boltzmann formula to hold, with an energy function that depends on the chemical potential as well. It should also be noted that temperature must equalize in any situation where the chemical potential equalizes, since it is impossible for systems to exchange molecules without also exchanging energy.

5) This algorithm, where we only choose one pixel at a time, is called Glauber dynamics. There are multiple ways to parallelize it, but the method chosen in this video is to only update each pixel with a low probability at each call of the compute shader, to avoid (frequently) updating two neighboring pixels simultaneously. One could also alternate between updating the even or odd pixels at each step.

6) By "zooming out," what is meant, technically, is a renormalization operation whereby each 3x3 grid of pixels is replaced by just one pixel, which has a molecule exactly when 5 or more of the pixels in the 3x3 grid have molecules. But that is somewhat tough to simulate as the size has to increase by 9 at each step, so this video cheats a bit by just zooming out on a large simulation.

7) The "interesting behavior" referenced here is the fact that the interface between the up and down sections of the picture can be rescaled (as the size of the model goes to infinity) in a way that yields a universal random curve called Schramm-Loewner evolution. This family of random curves shows up in many seemingly-unrelated places in statistical mechanics and probability.

This video made use of the following Creative Commons material:

https://commons.m.wikimedia.org/wiki/File:How_to_(nearly)_instantly_freeze_a_bottle_of_sparkling_water.ogv

https://commons.m.wikimedia.org/wiki/File:Blue_rotating_brain_animation.mpg

https://commons.m.wikimedia.org/wiki/File:7_Tesla_MRI_of_the_ex_vivo_human_brain_at_100_micron_resolution_(100_micron_MRI_acquired_FA25_coronal).webm

https://commons.m.wikimedia.org/wiki/File:Axonal_nerve_fibers_in_a_brain_-_the_neural_network_that_is_us.webm

3Blue1Brown

A comprehensive exploration of phase transitions through the liquid vapor model, introducing the Boltzmann distribution and its role in statistical mechanics. The video demonstrates how temperature and chemical potential control phase behavior, explains the mathematical foundations behind phase transitions, and shows how simplified models can reproduce complex real-world phenomena like the phase diagram of water. Key concepts include the competition between energy minimization and entropy maximization, the derivation of the Boltzmann formula, and implementation of Markov Chain Monte Carlo sampling methods.

Simulating Phase Change | Guest video by Vilas Winstein